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Dynamic investigation of nonlinear free vibration of circular plates resting on Winkler and Pasternak foundations

S. A. Salawu, M. G. Sobamowo, O. M. Sadiq

Abstract


Dynamic behavior of circular plate in contact with fluid and resting on two-parameter elastic foundations are investigated. This research is on the dynamic investigation of nonlinear free vibration of circular plates in contact with fluid also resting on Winkler and Pasternak foundations under different boundary conditions. The governing equation is transformed into the Duffing equation using the Galerkin method for the nonlinear analysis of the circular plates resting on Winkler and Pasternak foundations. However, the analytical solutions are obtained using differential transformation method. The developed analytical solutions obtained are used to investigate the influence of Pasternak and Winkler foundations, fluid, boundary conditions, radial and circumferential stress on the dynamic behavior of circular plates. Also, the accuracy of the analytical solutions obtained is verified with numerical results as reported in the literature. From the results, it is observed that the natural frequency of the circular plate increases with increases in the elastic foundation. The natural frequency decreases when the circular plate is in contact with water. The nonlinear vibration frequency ratio increases with an increase in nonlinear foundation stiffness. The node and antinodes of the mode shapes are affected by radial and circumferential stress. The lowest frequency ratio is observed at the clamped edge of the boundary condition. The novelty of the study includes the way the singularity value is handled. It is expected that the present study will contribute to the existing knowledge in the field of classical vibration.

Cite this Article: S.A. Salawu, M.G. Sobamowo, O.M. Sadiq. Dynamic Investigation of Nonlinear Free Vibration of Circular Plates Resting on Winkler and Pasternak Foundations. International Journal of Mechanical Handling and Automation. 2019; 5(2): 19–42p.


Keywords


Nonlinear free vibration; natural frequency; fluid; Winkler and Pasternak foundations; Differential transform method

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References


Soni S, Jain NK, PV Joshi. Analytical modeling for non-linear vibration analysis of functionally graded plate submerged in fluid. Indian Journal of Science Research. 2017; 4(2): 229–236.

Andrea RD Silva, Ricardo AM, Silveira, Paulo B Gonçalves. Numerical methods for analysis of plates on tensionless elastic foundations. International Journal of Solids and Structures-Pergamon. 2001; 38: 2083–2100.

H Lamb. On the vibrations of an elastic plate in contact with water. Proceedings of the Royal Society of London. Series A. vol. 98., 2016. pp. 205–216.

LB Raoa, CK Raob. Vibrations of circular plates resting on elastic foundation with elastically restrained edge against translation. The Journal of Engineering Research (TJER) 2018; 15(1): 14–25.

Z Xiao-Jing. An exact solution of Karman’s equations of rigid clamped circular plate and shallow spherical shell under a concentrated load. Applied Mathematics and Mechanics (English Edition). 1987; 8(11): 1057–1068.

D Haojiang, X Rongqiao, C Weiqiu. Exact solutions for free vibration of transversely isotropic piezoelectric circular plates. Acta Mechanica Sinica (English Series). The Chinese Society of Theoretical and Applied Mechanics Chinese Journal of Mechanics Press. Beijing, China Allerton Press, INC., New York, USA. 2000; 16(2): 141–147.

S Kumar, P Prashar. Energy method to analysis the vibration of orthotropic circular plate with bi-dimensional thickness variation. International Journal of Applied Research. 2015; 1(12): 829–835.

AM Siddiqul, T Haroon, S Bhatti, AR Ansari. A comparison of the Adomian and Homotopy perturbation methods in solving the problem of squeezing flow between two circular plates. Mathematical Modeling and Analysis. 2011; 15(4): 491–504.

Y Rostamiyan, A Fereidoon, MR Davoudabadi, H Yag. Analytical approach to investigation of deflection of circular plate under uniform load by Homotopy perturbation method. Mathematical and Computational Applications. 2010; 15(5): 816–821.

Y Yin-shan, C Temuer. Application of the Homotopy perturbation method for the large deflection problem of a circular plate. Applied Mathematical Modelling. 2015; 39: 1308–1316.

JK Zhou. Differential Transformation and Its Applications for Electrical Circuits. Wuhan, China: Huazhong University Press; 1986.

HS Yalcin, A Arikoglu, I Ozkol. Free vibration analysis of circular plates by differential transformation method. Applied Mathematics and Computation. 2009; 212(2): 377–386. doi:10.1016/j.amc.2009.02.032

M Shariyat, MM Alipour. Differential transform vibration and modal stress analyses of circular plates made of two-directional functionally graded materials resting on elastic foundations. Arch Applied Mechanism-Springer-Verlag. 2011; 81(9): 1289–1306.

Sobamowo MG. Nonlinear thermal and flow-induced vibration analysis of fluid-conveying carbon nanotube resting on Winkler and Pasternak foundations. Thermal Science and Engineering Progress. 2017; 4: 133–149. http//doi.org.//10.1016/j.isep.2017.08.005.

Sadiq OM, Sobamowo MG, Salawu SA, Yinusa AA. Dynamic response analysis of thin isotropic rectangular plates submerged in fluid using a new hybrid method. International Journal of Fracture and Damage Mechanics. 2019; 5(1): 32–59.

Kwak MK. Vibration of circular plate in contact with water. Transaction of ASME. 1991; 58: 480–483.

SA Eftekhari. Pressure-based and potential-based differential quadrature procedures for free vibration of circular plates in contact with fluid. Latin American Journal of Solids and Structures. 2016; 13: 610–631.

Y Kerboua, AA Lakis, Thomas M, L Marcouiller. Vibration analysis of rectangular plates coupled with fluid. Applied Mathematical Model. 2008; 32(12): 2570–2586. doi:10.1016/j.apm.2007.09.004

TY Wu, GR Liu. Free vibration analysis of circular plates using generalized differential quadrature rule. International Journal of Solids and Structure. 2001; 38(44): 7967–7980.

PC Dumir. Non-linear vibration and post-buckling of orthotropic thin circular plates on elastic foundations. Applied Acoustics. 1986; 19(6): 401–419.

S Chakraverty. Vibration of Plates. London: CRC Press Taylor & Francis Group; 2009. p. 44.

Amabili M, Frosali G, Kwak MK. Free vibrations of annular plates coupled with fluids. Journal of Sound and Vibration. 1996; 191(5): 825–846.

Jhung MJ, Choi JH, Ryu YH. Free vibration of a circular plate with eccentric hole submerged in fluid. Nuclear Engineering and Technology. 2009; 41(3): 355–364.

AW Leissa. Vibration of Plates. Washington DC: National Aeronautics and Space Administration, (NASA SP.160); 1969.

Thompson JMT, Stewart HB. Nonlinear dynamics and chaos. United States: John Wiley & Sons; 2002.


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